Spectral Theory of Quaternionic Operators

In this thesis, we concentrate on the spectral theory of quaternionic operators. First we prove the spectral theorem for quaternionic compact normal operators. Later we concentrate on the spectral theorem for quaternionic normal operators (not necessarily bounded). Here we prove the integral form as well as the multi- plication form. Then we establish the existence of the polar decomposition. Finally, we consider the functional calculus for quaternionic operators. This thesis contains six chapters. In Chapter 1 , we present some preliminaries of quaternionic Hilbert spaces, quaternionic linear operators and the motivation of this theory from the view point of quaternionic quantum mechanics. Also we recall some of the basic results from the literature which are useful for the subsequent chapters. Chapter 2 deals with the spectral theorem for quaternionic compact normal operators. We prove two versions, namely the series representation as well as the resolution of identity form. This result is available in the literature but we prove this by using simultaneous diagonalization process. The resolution of identity ver- sion is proved by reducing the problem to the complex case. In the process we prove some of interesting results on spherical spectrum of compact operators and the singular value decomposition. In Chapter 3 , we continue the study of spectral theorem in quaternionic Hilbert spaces. We prove the multiplication form of spectral theorem for quaternionic normal operators. In this process, we prove that every complex Hilbert space is a slice Hilbert space of some quaternionic Hilbert space. Also we prove that there exists an anti self-adjoint and unitary operator which commutes with the given densely defined, closed quaternionic normal operator. In Chapter 4 , we give an approach to continuous functional calculus for quater- viii nionic normal operators. Also deduce the integral representation of unbounded normal operators in quaternionic Hilbert spaces. In Chapter 5 , we extended the idea of functional calculus to the class of Borel functions as well as L ∞ - functions. In Chapter 6 , we prove the existence of polar decomposition of densely defined, closed quaternionic right linear operators. We give a necessary and sufficient con- dition for the uniqueness of polar decomposition.